where \(\mathbf {X}\)  is constrained to be a one-to-one mapping between the sets of nodes \(\mathbf {n}^{A}_{i}\)  and \(\mathbf {n}^{B}_{i}\) , and \(\text{vec}(\mathbf {X})\)  denotes the vectorisation of the correspondence matrix. We formulate both node- and edge-similarity metrics for the definition of the affinity matrix \(\mathbf {K}\) , by adopting the matrix factorisation as in [1]}, i.e. \(\mathbf {K}_{\mathbf {n}^{\text{\textit {AB}}}}\)  and \(\mathbf {K}_{\mathbf {e}^{\text{\textit {AB}}}}\)  respectively. In detail, we define
\(\vspace{-6.0pt}\mathbf {K}_{\mathbf {n}^{\text{\textit {AB}}}} = e^{-\left( \alpha _{1} \frac{\mathbf {C}^{\text{\textit {AB}}}}{\sigma _{\mathbf {C}}} ~+~ \alpha _{2} \frac{\mathbf {D}^{\text{\textit {AB}}}}{\sigma _{\mathbf {D}}} \right)} ~~~~ \text{with} ~~ \alpha _{1} + \alpha _{2} = 1,~~~ \text{and}\\\mathbf {K}_{\mathbf {e}^{\text{\textit {AB}}}} = e^{-\left( \beta _{1} \frac{\mathbf {P}^{\text{\textit {AB}}}}{\sigma _{\mathbf {P}}} ~+~ \beta _{2} \frac{\mathbf {L}^{\text{\textit {AB}}}}{\sigma _{\mathbf {L}}} ~+~ \beta _{3} \frac{\mathbf {U}^{\text{\textit {AB}}}}{\sigma _{\mathbf {U}}} \right)} ~~~~ \text{with} ~~ \beta _{1} + \beta _{2} + \beta _{3} = 1,\vspace{-6.0pt}\) 
